How to add an Image to a plane that has a boolean modifier applied?
I'm not 100% sure it's clear from the questions what your aim is.

Asuming you'd like more control over how images are displayed on the final object I'd go with one of two options:


Apply the boolean modifier and work with separate faces in Edit mode
Use the same material to both planes (bool operands) but have different UV mappings for them, so that you can use separate areas of your image to texture the final united plane.


In the example below I did this to both planes with the same image texture applied (same material). However, one plane has UV square on the left side of the image while the other plane has UV on the right side.

You can hide the second plane and the final object would still look the same.

You could do different things then:


Shrink one of the UVs to a pixel sized square and locate it over the needed color to have a mono-coloured part


OR


Scale UV by -1 to reverse the mirror effect you have on your screenshot.


Hope this helps.